The Equality of 3-manifold Invariants

The invariants of 3-manifolds defined by Kuperberg for involutory Hopf algebras and those defined by the authors for spherical Hopf algebras are the same for Hopf algebras on which they are both defined. Introduction The purpose of this paper is to compare two previously defined invariants of 3-manifolds. Let A be a finite-dimensional Hopf algebra over a field F with antipode S. Then if S = 1 the Hopf algebra is said to be involutory. Let A be involutory and the dimension of A in the field F be not zero. Then it follows from [Larson and Radford 1987] that A is semisimple and cosemisimple. For each such Hopf algebra A, Kuperberg [1990] has defined an invariant of closed oriented 3-manifolds. For a manifold M , this invariant is denoted K(M). The present authors defined an invariant of closed oriented 3-manifolds for each such Hopf algebra A over an algebraically closed field [Barrett and Westbury 1993; proposition 6.8]. This invariant is denoted Z(M) for a manifold M and is called a state sum invariant. The result of this paper is Theorem. Let A be a finite dimensional involutory Hopf algebra over an algebraically closed field, with dimA 6= 0. Then K(M) = Z(M) dimA. for all M . This result implies the following relationship between the scope of the two invariants. The state sum invariants of Barrett and Westbury [1993] are defined for the more general notion of a finite semisimple spherical category of non-zero dimension. This generalises the notion of the category of representations of a semisimple Hopf algebra. As we showed in that paper, examples can be constructed from Hopf algebras which are not themselves semisimple, such as the quantised universal enveloping algebras. For these non-semisimple Hopf algebras it is apparently The hypothesis there that the field has characteristic zero can be replaced by the hypothesis that the algebra has non-zero dimension.